Related papers: $L^2$-stableness for solution to linearized KdV eq…
We prove global well-posedness of the subcritical generalized Korteweg-de Vries equation (the mKdV and the gKdV with quartic power of nonlinearity) subject to an additive random perturbation. More precisely, we prove that if the driving…
For quantum mechanics of a charged particle in a classical external electromagnetic field, there is an apparent puzzle that the matrix element of the canonical momentum and Hamiltonian operators is gauge dependent. A resolution to this…
In this paper, we study a linearized two-dimensional Euler equation. This equation decouples into infinitely many invariant subsystems. Each invariant subsystem is shown to be a linear Hamiltonian system of infinite dimensions. Another…
We consider the stability and instability of periodic travling waves for Korteweg-de Vries type equations with fractional dispersion and other nonlinear dispersive equations. We establish that a constrained minimizer for the related…
We propose a new family of complex PT-symmetric extensions of the Korteweg-de Vries equation. The deformed equations can be associated to a sequence of non-Hermitian Hamiltonians. The first charges related to the conservation of mass,…
We provide a detailed study of the dynamics obtained by linearizing the Korteweg-de Vries equation about one of its periodic traveling waves, a cnoidal wave. In a suitable sense, linearly analogous to space-modulated stability, we prove…
We prove the asymptotic stability in the energy space of non-zero speed solitons for the one-dimensional Landau-Lifshitz equation with an easy-plane anisotropy. More precisely, we show that any solution corresponding to an initial datum…
Linear Hamiltonian systems with time-dependent coefficients are of importance to nonlinear Hamiltonian systems, accelerator physics, plasma physics, and quantum physics. It is shown that the solution map of a linear Hamiltonian system with…
We establish the conditional asymptotic stability in a local energy norm of the unstable soliton for the one-dimensional quadratic Klein-Gordon equation under even perturbations. A key feature of the problem is the positive gap eigenvalue…
We consider some basic problems associated with quantum mechanics of systems having a time-dependent Hilbert space. We provide a consistent treatment of these systems and address the possibility of describing them in terms of a…
A variable-coefficient forced Korteweg-de Vries equation with spacial inhomogeneity is investigated in this paper. Under constraints, this equation is transformed into its bilinear form, and multi-soliton solutions are derived. Effects of…
Stationary solutions on a bounded interval for an initial-boundary value problem to Korteweg--de~Vries and modified Korteweg--de~Vries equation (for the last one both in focusing and defocusing cases) are constructed. The method of the…
In a series of recent works by Demirkaya et al. stability analysis for the static kink solutions to the 1D continuous and discrete Klein-Gordon equations with a $\mathcal{PT}$-symmetric perturbation has been analysed. We consider the linear…
Stability of soliton families in one-dimensional nonlinear Schroedinger equations with non-parity-time (PT)-symmetric complex potentials is investigated numerically. It is shown that these solitons can be linearly stable in a wide range of…
We consider periodic solutions to equations of Korteweg-Devries type. While the stability theory for periodic waves has received much some attention the theory is much less developed than the analogous theory for solitary wave stability,…
The ordinary Landau problem consists of describing a charged particle in time-independent magnetic field. In the present case the problem is generalized onto time-dependent uniform electric fields with time-dependent mass and harmonic…
We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its…
The Lindblad equation is widely employed in studies of Markovian quantum open systems. Here, firstly, a simple result is presented on the time evolution of the non Neumann entropy under the Lindblad equation, which enables one to examine if…
In this paper we present a set of results on the symmetries of the lattice Schwarzian Korteweg-de Vries (lSKdV) equation. We construct the Lie point symmetries and, using its associated spectral problem, an infinite sequence of generalized…
Harmonic oscillator in noncommutative two dimensional lattice are investigated. Using the properties of non-differential calculus and its applications to quantum mechanics, we provide the eigenvalues and eigenfunctions of the corresponding…